MathDB

Problem 3

Part of 2023 VN Math Olympiad For High School Students

Problems(2)

<APB+ <MPC=180

Source: 2023 Viet Nam math olympiad for high school students D1 P3

3/25/2023
Given a triangle ABCABC isosceles at A.A. A point PP lying inside the triangle such that ∠PBC=∠PCA\angle PBC=\angle PCA and let MM be the midpoint of BC.BC.
Prove that: ∠APB+∠MPC=180∘.\angle APB+ \angle MPC =180^{\circ}.
geometry
Eisenstein's criterion

Source: 2023 Viet Nam math olympiad for high school students D2 P3

3/25/2023
Given a polynomial with integer coefficents with degree n>0:n>0:P(x)=anxn+...+a1x+a0.P(x)=a_nx^n+...+a_1x+a_0. Assume that there exists a prime number pp satisfying these conditions: i) p∣aip|a_i for all 0≤i<n,0\le i<n, ii) p∤an,p\nmid a_n, iii) p2∤a0.p^2\nmid a_0. Prove that P(x)P(x) is irreducible in Z[x].\mathbb{Z}[x].
algebranumber theoryprime numbers